Understanding Extreme Numbers

This is not just an academic idea for programmers, it is something that you will need to understand to get to the highest levels of engineering. In my career, I have cared about how many nanoseconds an operation takes, and I have optimized processes that required several hours to run. That's 9 orders of magnitude down from one second, and if for convenience we go with my hours-long process taking 10000 seconds (2.78 hours), that's four orders of magnitude up.

Obviously, there are people with far more experience than I on the upside; if you work with a process that will run on a 10,000 (CPU) node cluster that will run for 1,000,000 seconds (11.6 days) that's 10 orders of magnitude up from 1CPU for 1sec... and they're actually still quite likely to care about nanosecond level performance, too! Which means that someone working in this situation is spanning 19 orders of magnitude in their day-to-day work.

19 orders of magnitude down from 1 meter is about five orders of magnitude smaller than a hydrogen atom. 19 orders of magnitude up from one meter is about 1000 light-years. It's a pretty big span.

There are not a lot of engineering disciplines, or really, disciplines in general, where spanning this many orders of magnitude in a single job is possible. When it comes time to start optimizing things, you need to know where you stand in the world to make correct decisions: https://stackoverflow.com/questions/4087280/approximate-cost...

> That means they correctly distinguish between numbers within the millions or billions, but assume that “million,” “billion,” and “trillion” are equally spaced on a number line.

Maybe normal people are good at log scales! Our number system makes log_10 pretty natural, and our naming system makes log_1000 somewhat automatic.

I think it's only fair to point out that log scales are how scientists and engineers understand extreme numbers; nobody really has an "intuitive" understanding of 10^36 or Graham's number.

I can feel this myself; it's easy to think of 1 million as a thousand squared and feel like it's tractable, it's easy to think of a billion as a thousand cubed and feel like it's relatively small. But when I actually plot or visualize that many things, especially if the individuals have variation, I can see that thinking of it in exponents of a thousand is misleading, and that it's hard to get any sense of how many individuals are there and of the complexity of variations and sub-groups that are possible.

> But about 35 percent of people in the study used what the authors call a “segmented linear heuristic.” That means they correctly distinguish between numbers within the millions or billions, but assume that “million,” “billion,” and “trillion” are equally spaced on a number line.

That kind of logarithmic thinking is a reason why I believe people don't react more strongly when they hear that Jeff Bezos is worth 100 billion dollars.

The wealth of the super rich is so astronomical that people can't make sense of it. It doesn't help to say he's 100,000 times a millionaire, because even if we more or less know how wealthy a millionaire is, we can't imagine what multiplied by 100,000 means.

Background: I used to be an physicist (PhD) and graduated as an engineer (applied physics). I also actively promote science with children. I am interested in the interpretation of numbers and quantities for many years.

I do not like this article because it fails to

- show that we do not "feel" numbers larger than a few

- and that we have a tendency to think in orders of magnitude (5, a few, lots, plenty of many).

To start with: "many treated 980 million and 2 billion as nearly identical". They were right, 980 million (0.98 billion) is nearly identical to 2 billion when you look at orders of magnitude.

Missing this fact is one of the big problems we have at school, namely the incapacity to estimate. We fetish exact numbers (3.14) instead of telling that pi is 3, pi squared is 10 and g is 10. This way, children can understand orders of magnitude and we do not end with (real case when I was teaching) a skydiver who lands at 573.426625432423 km/h. I cannot deny that the answer is precise to the upth-st digit.

Quantities such as "a million" do not make sense outside of the context. In context they just become a number, not a quantity. For instance what is the size of a hair multiplied by a million? Or a glass? I have no idea. Sure, I can estimate that a glass is 5 cm = 5 x 10^-2 m, which makes it 5 x 10^-2 x 10^6 = 5 x 10^4 m. 50 km. But I had no idea before because I don't know what "a million times" means outside of the numbers.

We are good at estimating small orders of magnitude (from experience, not more than 3) and that's all. The rest is calculus and this is easy. But also detached from reality, a scientist does not ponder on 10^9. This is jst 10^9, a number.

Not claiming to be exceptional, just noting that a perspective I developed seems less common than I expected...

I think I've just had a cognitive bias of my own revealed, and found a source of confusion in conversation... I guess I've been sucking up any and all science that interests me since I could read, I have no trouble perceiving scale differences... Planck's constant, universe-scale light-year measurements, Ackermann function values, they are all on a very long and flexible scale of powers of ten, and that whole scale is mostly intuitive. I sometimes even convert everyday numbers into E notation so I know I'm scaling things right.

Thanks to existence of the short and long scales I have no idea what people mean when they say "a billion" or something like that today in the globalized Internet-connected world. Please avoid using such ambiguous words at all costs, use scientific notation instead.

I suggest the article is missing a key point.

> Of course, our ancient human ancestors didn’t live among billions of people, or incur trillions of dollars of debt. The orders of magnitude in their immediate surroundings were limited to what they could experience firsthand.

A square kilometer might have a billion trees. With how many leaves? One person's year might have a billion heartbeats. A pre-industrial farmer might tend a million plants per year. A sack of rice might have a million grains.

> It’s not surprising that we can intuitively visualize a 6-inch or 6-mile distance,

Asking people how tall a skyscraper is, I get answers like "a mile?". And even for horizontal miles, who is this "we" who can "intuitively visualize" miles - people who drive?

> Things that are so far removed from our daily experience [...] are inherently hard to understand,

People are regularly innumerate and clueless about things that are part of their daily experience. Far remove is not a necessary condition. Nor is it a sufficient one - experience need not be direct. How often do kids see real horses? Or elephants? Or dinosaurs?

So I suggest the article is missing a key point.

Large quantities, large differences in scale, are inherent in the universe. It's what makes estimation and approximation possible. People simplify with units, landmarks and anchors, and rules of thumb. With experience and stories. But education, and press, leave people largely innumerate, making almost no effort to teach the needed landmarks. In science, and history, and civics, and finance. The main exceptions being a few units of time, and of currency, which are taught down to K.

Think of those little lists in introductory physics. For some measure (eg, Volts), here are a few values. Usually poorly chosen. And never built upon, into a feel for reasonable values.

And then we get science folks saying nanometers are unimaginably small. And journalists that a trillion dollars is unimaginably large.

Yes, people are largely innumerate. But we collectively make almost no effort to change that.

http://bionumbers.hms.harvard.edu/ started not from the movement to teach introductory biology more quantitatively, but out a graduate student's realization that they had no quantitative feel for their own field. A characterization repeatedly made by physicists about their own doctoral candidates. How many people here recognize a Newton-meter of torque, as being the reopening a 2L soda bottle?

So a takeaway shouldn't be researchers doing something novel, to address a new modern problem. But rather their getting around to doing something long known and established, to address a long standing problem. A problem long, pervasively, unnecessarily, and abjectly neglected.